Lemma 2.1.7.

Let and be unital -algebras, and let be a surjective (And hence unit preserving) star-homomorphism.

  1. For each , there exists such that
    Where is the map induced by .
  2. If is a unitary element in , and if there is a unitary element such that , then belongs to
    in other words lifts to a unitary element in .

Proof of 1:
A unital -homomorphism is continuous and maps unitaries to unitaries. Therefore .
Conversely, if then by Proposition 2.1.6. for some self-adjoint elements in . Since is surjective, there are elements such that Put
Then and
set
Then belongs to by Proposition 2.1.6. and .
Reflection:
The first containment follows from continuity of the map.
The second relies on surjectivity, we have this decomposition of elements in
into products of exponentials of self-adjoint elements, we can find elements who map onto them, we then. take the real part of each of the ’s, this gives us self-adjoint elements we need in order to represent the pre image as living in (A).

Proof 2:
Follows immediately from 1. and Lemma 2.1.5 (Whitehead).
[Insert detailed rumination]

Proof 3:
If , then belongs to , and hence for some by 1. Now as desired.